Visualization and Assessment of Copula Symmetry

Visualization and assessment of copula structures are crucial for accurately understanding and modeling the dependencies in multivariate data analysis. In this article, we introduce an innovative method that employs functional boxplots and rank-based testing procedures to evaluate copula symmetries. This approach is specifically designed to assess key characteristics such as reflection symmetry, radial symmetry, and joint symmetry. We first construct test functions for each specific property and then investigate the asymptotic properties of their empirical estimators. We demonstrate that the functional boxplot of these sample test functions serves as an informative visualization tool of a given copula structure, effectively measuring the departure from zero of the test function. Furthermore, we introduce a nonparametric testing procedure to assess the significance of deviations from symmetry, ensuring the accuracy and reliability of our visualization method. Through extensive simulation studies involving various copula models, we demonstrate the effectiveness of our testing approach. Finally, we apply our visualization and testing techniques to three real-world datasets: a nutritional habits survey with five variables, stock price data for the five top companies in the NASDAQ-100 stock index, and two major stock indices, the US S&P500 and German DAX. Supplementary materials for this article are available online.